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The paper is concerned with the Morse equation for flows in a representation of a compact Lie group. As a consequence of this equation we give a relationship between the equivariant Conley index of an isolated invariant set of the flow given by .x = −∇f(x) and the gradient equivariant degree of ∇f. Some multiplicity results are also presented.

A special case of G-equivariant degree is defined, where G = ℤ₂, and the action is determined by an involution $T:{ℝ}^p\oplus {ℝ}^q\to {ℝ}^p\oplus {ℝ}^q$ given by T(u,v) = (u,-v). The presented construction is self-contained. It is also shown that two T-equivariant gradient maps $f,g:(ℝⁿ,S^{n-1})\to (ℝⁿ,ℝⁿ\setminus 0)$ are T-homotopic iff they are gradient T-homotopic. This is an equivariant generalization of the result due to Parusiński.